How to align vertex to circle
I think you were along exactly the right track in using the 'space' command of the loop tools add-on; even spacing really is the best way to get the smoothest curve.

The reason that the 'space' command is distorting some of the edges is that there are more edges on one side of the arch than the other, so some edges have to be moved a long way before the arch is evenly spaced. Having a similar number of edges going into each side of the arch will help with this. 

I would try to either add a few extra loops into one side of the arch (marked in green in the screenshot below) or remove some loops, if possible, from the right side (marked in red):



Ideally, you would do a mixture of the two to even up the arch. Once this is done the 'space' command will not have such an extreme effect on the surrounding edges. I understand that adding more loops in will not seem ideal as you are probably (and rightly) keeping the topology as simple as possible, but in my experience, this is the right way to go.